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DAP AN tai eer! ale vy MAN MOUS) ited eg ‘Sty 4 phy ah et phere eens balls at , : ke | ’ Pay , ‘ ’ My m ) "y i, ‘ eh ne be a mies Bie te a at cae ries alae Hl i) uy ‘ Aan i q i j ysl 3 ¥it i 4 7 hy : iy ‘ ex 4 : : i : r ) i ” ities + by * Sal Al \ } as! fe 4) ’ \ Aa o's #2 ’ 1) iS ee 4 ? PP 8 Fit oe ie Hfieh iy! ‘ Pe a eke cl ” hee: Mi) fy } wii alah . + Tehri Witte a 3 que Const LUO esate’ anti bas hd ee eit ' ; r es ni vir br itr ene +H Hae ‘ys Ky ' 4 “4 i i yb Ha nena pened ita het ; i at iy ij ie oe Aesth ut 4 ae bh fy eae , SaaS r A , | haa ‘ ty Me red een at if s ’ ' tt a H ! ’ y Un siuthoye . i ; ty : ¥ ‘ ; ' Pt} “4 jiseoh ins ey \ Penta i - Hatt : , Si« * ‘ sete se , dag ' ‘ ated © f 1G Ne ii i! 8, bile Hf ee | , et ee a a Ah . eM ihe 1, ; i ub tate wy lean y a MuIES Sener e ae nals te oa he & 4 ‘ 7 : ¥ Dye rd cd om ’ Sian t : ie pd pl “a oe a 4 : om See eh : j aba fh, & Mh, ” ' ae wipiieen ret et va hy ‘te Tel teats miei Herta whet | et Taal tthe oy ae ; » ‘ alu a ye! ya ; Het ht i " d ah ! »' ‘ +f oven! j ' ue ala Ms ine) ee itt. rated mist Dass 16-18} s io } ne Svea! i tie vel ary th 63 ty a hye tk tbe Bs sai mt mt er) ; ; ‘ ‘iad Mitte htab? wot) poate Heh hs Ad al t¢ i 0 ee ¢ arly ny ty ‘ee aN P ni he reir +b; the id anche mihterielibe ibe ys ) ¢ J f hd qe Di ‘ iM? sekt A ihrer Aurea gaedy rh ee Ne ae Aa inl ee Live reat pa ataetowe hs Heit « au ‘ ne f 4 t pe! i ‘ ; ‘ ' wit y rs Bit bre de he hgh (1 baie pi 1 &y¥ . t Ba! ij : 554 POONA AINA YILY IDRIDRIAOWONIONOVONIONIONOIO’ DROOAIN Mme aaa aw MMAIMMAMMBBacAawK Q SMM MM MarBMUAMBwxwOKwM GOVERNMENT PRENTING OFFICE 11—8625 SREP Se a aie = 446 AN LONDON, EDINBURGH, «ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE, | : CONDUCTED BY ¥ SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.R:S. . SIR JOSEPH JOHN THOMSON, M.A., So.D., LL.D., F.RS .% JOHN JOLY, M.A., D.Sc, F.R:S., F.G.S. GEORGE CAREY FOSTER, B.A., LL.D., F.R.S. AND WILLIAM FRANCIS, F.LS. y / f ‘“‘ Nec aranearum sane textus ideo melior quia ex se-fila gignunt, nec noster vilior quia ex alienis libamus ut apes.’”’ Just. Ips. Polit. lib. i. cap. 1. Not. VOL. XXVIII.—SIXTH SERIES. JULY—DECEMBER 1914. irae 3RAARS . Bl ae \ DEC 171914 rap % MCA NE Sa ae ~ PATENT aA LONDON: TAYLOR AND KRANCIS, RED LION COURT. FLEET STREET, SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND UO., LD. SMITH AND SON, GLASGOW :— HODGES, FIGGIS, AND CO., DUBLIN; VEUVE J. BOYVHAU, PARIS ;—AND ASHER AND CO., BERLIN. ¥ “‘Meditationis est perscrutari occulta; coutemplationis est admirari perspicua .... Admiratio generat questionem, questio investigationem, investigatio inventionem.”—Hugo de S. Victore. ——‘ Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ceelo, Quy micet igne Iris, superos quis conciat orbes Tam vario motu.” J. LB. Pinelli ad Mazonium. CONTENTS OF VOL. XXVIII. (SIXTH SERIES), NUMBER CLXIII.—JULY 1914. Prof. A. Gray: Notes on Hydrodynamics.—I. .......... Prof. A. Gray: Notes on Hydrodynamics.—I]. .......... Prof. W. M. Thornton on the Lost Pressure in Gaseous NORE Sai) S 2) sei vied wieteje esl! eyes die Wale oe haste Mr. J. Proudman on the Motion of Viscous Liquids in Mia a Salat sald o's) od x wip) sc vista cece he «oe oa e Dr. J. P. Dalton on a New Continuous-Balance Method of Comparing an Inductance with a Capacity ..,......... Dr. Herbert Edmeston Watson and Mr. Gostabehari Pal on the Radioactivity of the Rocks of the Kolar Gold-Fields . . Mr. Gilbert Cook on the Collapse of Short Tubes by External EEMEMBALGOD Ve eis pik vise ew a eR ee hans © hap Dr. I. J. Schwatt: Note on a Definite Integral .......... Dr. Max Planck on New Paths of Physical Knowledge Dr. Eva von Bahr on the Quantum-Theory and the Rotation- EST Tg Prof. J. 8. Townsend on the Potentials required to Maintain Currents between Coaxial Cylinders .................. Prof. J. W. Nicholson on Atomic Structure and the Spec- Ce ens ee ee ee eee Prof. W. C. McC. Lewis on the Relation of the Internal Pressure of a Liquid to its Dielectric Capacity and Permea- Ee ails, PRS diwatl-ce tien d' Qawe's sae wiad es ble oe Dr. W. Marshall Watts on the Spectra given by Carbon and some of its Compounds; and, in particular, the ‘ Swan” ER EE ee eee en ee ee Mr. Allan Ferguson on the Shape of the Capillary Surface inside a Tube of Small Radius, with other Allied Problems. Prof. W. M. Hicks on High-frequency Spectra and the Roe! 5S x iy, 4 we sisi ds ye sw NS ee eee ee ee Prof. J. R. Rydberg on the Ordinals of the Elements and meee riiehetrequeney SpectPa 6. cele ed ewes ase Mr. Allan Ferguson on the Forces acting on a Solid Sphere in contact with a Liquid Surface (II.) ................ Notices respecting New Books :— Mr. Frederick Soddy’s The Chemistry of the Radio- (a NAURSARRL ge GET OR a eA Sg ie aA ee 90 RN Pe 1V CONTENTS OF VOL. XXVIII.—-SIXTH SERIES. Proceediny's of the Geological Society :— Lady McRobert on Acid and Intermediate Intrusions in the! Neighbourhood’ of Melrose”. 0.4 22.2 2.92 2 eae Prof. J. W. Judd on the geology of Rockall .......... Dr. H. 8. Washington on the Composition of Rockallite. Prof. J. W. Gregory on the Evolution of the Essex Raiver-System. 55 Larter an ae ie Mr. J. B. Scrivenor on the Topaz-bearing Rocks of Gunong Bakau (Federated Malay States) .......... NUMBER CLXIV.—AUGUST. Lord Rayleigh on the Equilibrium of Revolving Liquid under Capillary Monee 10 22240 5 A MN FS LEA Dr. Charles Sheard on the Positive Ionization-from Heated Platina (Oe eS a SEAS OSs 2 ae Messrs. C. E. Magnusson and H. C. Stevens. on Visual Sen- sations caused by a Magnetic Field. (Plates III. & IV.) . Dr. Albert C. Crehore on the Theory of the String Galvano- meter of Birtthoven! 2's. ea Re eS ae Prof. D. N. Mallik on the Dynamical Theory of Diffraction . Mr. P. J. Edmunds on the Discharge of Electricity from Page 156 157 158 160 Points Se a Sey ee eg eee ie ek 2 ee 234 Dr. Frank Horton on the Action of a Wehnelt Cathode .... 244 Dr. C. V. Burton on the Possible Dependence of Gravitational Attraction on Chemical Composition, and the Fluctuations of the Moon’s Longitude which might result therefrom Mr. 8. 8S. Richardson on Polarizing Prisms for the Ultra- Vidleb ove tien ead seek e CROLL La ea Pr Sir Ernest Rutherford and Dr. E. N. da C. Andrade on the Spectrum of the Penetrating y Rays from Radium B and Radmmo@:” (Plate: Vaye oo 8 0 AU. Oe é Mr. W. F. Rawlinson on the X-ray Spectrum of Nickel .... £ Messrs. H. Robinson and W. F. Rawlinson on the Magnetic Spectrum of the § Rays excited in Metals by Soft X Rays Sir Ernest Rutherford and Messrs. H. Robinson and W. F. Rawlinson on the Spectrum of the 6 Rays excited byjy Rays. Dr. Norman Campbell on the Ionization of Platinum by Catone itanyis eye oS Lk SR SR Dr. G. Bruhat on the Theories of the Rotational Optical WCHARF GR eae Sse eee eat elmnih eee em eee an NUMBER CLXV.—SEPTEMBER.’ Sir Ernest Rutherford on the Connexion between the 6 and oy TRA) BO CERD UN riesig «s5feis)a ce +e eye seagee te shel alana ee Sir Ernest Rutherford: Radium Constants on the Inter- nabional SeanGatd oes 2... s. Tea eases ae olan eee Messrs. H. G. J. Moseley and H. Robinson on the Number _of Ions produced by the 6 and y Radiations from Radium. 320 327 CONTENTS OF VOL. XXVIII.—SIXTH SERIES. Prof. A. Ll. Hughes on the Contact Difference of Potential of MC Mieee Peel Ne zie siete ey eames aicle ws Toa swt Sed 6 v Page Prof. L. L. Campbell on Disintegration of the Aluminium — MEARE 5S Fetstot cae We ede Ale MRA oie nical Day Ute oS Mr. W. Lawrence Bragg on the Crystalline Structure of PEs Sk ene Soke. Ye Were we wine Biv vita Gu lod hee dile Prof. J. C. McLennan on the Absorption Spectrum of Zinc Sree Cr abe Vi.) as 3s Neth hie Wav iota’ alls sled. Qiaatee Mr. D. C. H. Florance on Secondary y Radiation 347 309 360 363 Mr. H. A. McTaggart on Electrification at Liquid-Gas Surfaces 367 Mr. L. Isserlis: The Application of Solid Hypergeometrical Series to Frequency Distributions in Space. (Plate VII.) . Mr. Allan Ferguson on the Surface-tensions of Liquids in esien with different (ases 4.) een ec shene selene Dr. T. Martin Lowry on an Oxidizable Variety of Nitrogen. MME NAD. olsun acyaje b gsa'le ] 25 072 "Y Sc0y =i Xo X10 tae P.615. In the denominator of equation (12), for (1+, 6°) read MV (1+7'5 Be’). P. 684, line 8, the first integral inside the bracket should be ry 1 P,idz : Pdr ee) sien, a \ (+ h0*)? re \, (I+ #2") e ‘This is printed correctly in the majority of copies of the November nunber, but in some of the later “pulls” the ® appears to have been broken off.— W. F. | THE LONDON, EDINBURGH ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] FUL ¥ A914, I. Notes on Hydrodynamics. By Professor A. Gray, F.R.S.* I. Equations of Motion for any Axes. Theorems of Cir- culation. Proof of Constancy of Moment of Vortea- Filament. 1, | T time ¢ the equation of motion along a stream-line in a perfect fluid is Be Oa CONN On PL ge Be THs =} | Samer en 3s. where p is a function of p. For the first two terms are the acceleration along the stream- line at the point considered, P say, and the third and fourth, with their signs changed, make up the force per unit volume due to the field of force for which the potential at P is V, and to the gradient of pressure Qp/Os. Integrating along the stream-line from a chosen initial point P, to P as a final point, we obtain fol] dp 4dast+4(g—g?)+V—V+)]—=0.. . (2 “ ot 7 Cis 1.) Edn By p ) PoP PP * Communicated by the Author. These Notes are founded on a paper published in the Trans, R.S. E. 1908, but the subject is presented in an improved manner, and much of the matter is entirely new. Phil. Mag. 8. 6. Vol. 28. No. 163. July 1914. B 2 Prof. A. Gray: Notes on Hydrodynamics. This is generally called Bernoulli’s theorem (Daniel Bernoulli, Hydrodynamica, 1738). 2. Now if instead of considering, Fig. 1. as in (1), the acceleration along Bic the stream-line, we take the acce- eee leration along a step ds! drawn from Ue P (fig. 1) in a direction making an y angle @ with the stepds drawn from ae the same point along the stream- line, we get as ies equation of motion in the new direction oe, 09 at 2 one ide Ot eel as p Os aN Rina =0. (3) If we write 9 19 we have, identically, Og 0d |. OVE hap og en ot oa tet. OS Bi a Haare ee. Subtracting from (3) we obtain fe) Os os Qs” But if w,,. denote the component of elemental rotation of the fluid about the normal (drawn outwards from the paper, see figure) to the plane of ds and ds' at P, we have, as we shall prove below, 096 ag 255! sin (pee — a $ so that (5) may be written § oe + 2@,.'9 sin 0= — OM, ee 3. By turning ds’ without altering 0, we can change the plane of ds and ds’ from that for which o,, is zero to that— inclined to the former at an angle 47—for which ogy sin 0 isa maximum. LHquation (6) thus shows how we can pass, in any direction, from the value of y at a point P on one stream-line to the value of y at an adjacent point P’ on another. Thus along any surface about the normals to which at every point there is zero rotation ss, the value of — Prof. A. Gray: Notes on Hydrodynamics. 3 x is, in the case of steady motion, constant from stream-line to stream-line. If there is no elemental rotation anywhere throughout any finite portion of the fluid in steady motion the value of y is the same for all stream-lines in that portion. 4. To prove that @,,, defined as above, has the value given by the equation / ee 25s! SID got _ Od (a) we may proceed thus. Consider the parallelogram (see figure) of which adjacent sides are PP’, PQ, that is ds’, ds. The average velocities of the fluid along the four sides PQ, QR, RP’, P’P are 1 09 09’ a ie ds, g+ 5 dst 3 Shas i 10g 07 i eS gl as ds+ 24s!) (1+5 $4 ds'), The first and third of these multiplied by ds, and the second and fourth multiplied by ds’, give a sum of products which is the circulation round the parallelogram.. The sum is (dq'/ds—dq/0s') ds ds’. In other words, if g, be the com- ponent of velocity along the boundary of the parallelogram at any element dc, we have for the parallelogram (o.ae=(§4 - of) ds ds’ ‘But if P” be a point within the parallelogram in the plane of ds and ds’, and p be the perpendicular distance of P" from the element of periphery dc, the angular velocity of a fluid particle at de about P” is g-/p. Thus we have Gai 5) (0g! Og {2 pdc= he pia! ;) ds ds. If «3 be the mean angular velocity about P” for the particles of fluid on the periphery at the instant considered we get, since | p de is twice the area of the parallelogram, and this is also 2 ds ds’ sin @, | iy |) OM) 2a, sin 0 = vias Thus (7) is proved. 4 Prof. A. Gray: Notes on Hydrodynamics. It will be noticed that this result is independent of the position of the point P’’ with respect to which the angular velocities of the particles have been taken. A small spherical portion of the fluid with its centre within the infinitesimal parallelogram, has a component of angular momentum about the normal to the plane of ds ds', of amount 8 17” Wss'; where 7 is the radius of the radius and p the density of the fluid. ‘Thus we call @,. the component of elemental angular velocity of the fluid about the normal considered. It is an affair of an element of the fluid mass, not of a particle. 5. It is easy to extend the process by which (7) has been established to show that the circulation round any closed path drawn in the fluid is equal to twice the surface integral of rotation of the fluid about the normals to the elements of any surface of which the path is the bounding edge, and indeed to prove Stokes’s theorem connecting the line integral of a directed quantity taken round the boundary of a surface, with integral of the curl of the same quantity taken over the surface. In what follows we shall assume the theorem of circulation (not, however, of the constancy of circulatiou for a closed path moving with the fluid). 6. We may interpret the result stated in (5) and (6) above in the following manner. First integrate along the path of which PP’ is an element from an initial point A toa final point B. We obtain from (6) Xn X= 2 V wg sin 6 ds’ — SE (8) . ie Pe ot ° e L} AB AB Now as q is the resultant velocity of the fluid at any point P on the path AB, g' (=q cos @) is the component at P along the tangent drawn there to AB, while gsin@ is the velocity with which each particle P on the path AB is being carried by the motion towards the right (see fig. 2) in the plane of the diagram. The product gsin @ds’ is therefore the rate at which an area of which ds’ is an element of boundary, and which is situated to the left of AB, is in- creasing (or if the area is situated to the right of AB, is diminishing) in consequence of the motion of ds’ as a whole, at right angles to itself, in the plane of the diagram. We see, therefore, that the first term on the left is twice the rate at which the surface integral of elemental rotation, taken over Prof. A. Gray: Notes on Hydrodynamies. 5 any area of which AB is part of the boundary, is changing in consequence of the fact that each element ds' of AB is being carried towards the right by the motion of the fluid. Fig. 3 shows the effect of this motion for a closed path of inte- gration. The area between two stream-line elements ds, and the two positions of the connecting ds’, is evidently ds ds! sin 0. The second term on the right is the rate at which as time passes the flow along changing apart from the motion” elements ds’. | If the pat ABbe soi ne BB) Kip a where ({) denotes integration eel the close 7. Now since the line integral fq ds' taken road the closed path is twice the surface-integral of elemental rotation about normals drawn to the elements into which any surface bounded by the path may be divided, if we calculate the whole change of rate of flow along AB due to the various causes, we shall obtain a result which, extended to the whole circuit, will give the exact rate of increase of the surface integral of elemental rotation for any surface of which the path of integration is the bounding edge. Now the surface integral is increasing at rate 2 (|) ose qsin 6 ds’, in consequence of the motion of the elements ds’ at right angles to themselves, and also at rate (f oe ds' in consequence of the variation of g' with time. There are two other causes of variaticn due to the motion. Hach element ds' is being displaced bodily in the direction of its length, and moreover the end of the element nearer B is undergoing displacement with respect to the end nearer A. It can be proved that each of these displacements gives a rate of increase of flow along the element of amount 6 Prof. A. Gray: Notes on Hydrodynamics. q0qg/ds'.ds'. Thus the whole rate of diminution of flow along the unclosed path AB is a {2h ow qsin 0 ds'+2 724 pds + + (SC ah at AB and so from (8) we get OF ae =i, is / OG 5 Og A x—x—2 (aS he = 24 oaiqsin Bd +2 ee + re AB AB AB AB (10) The left-hand side is thus the whole rate of diminution of flow along AB, and, since when extended to a complete circuital path, it gives the whole rate of change of the surface integral of spin about the normals, may be regarded also as twice the rate of diminution of this surface integral as depending on AB. Writing fq’ ds = 21 AB ws have from (10) when the value of y is inserted eg —1¢—(V+("? -37°), - fee A LEU MS) B B 07 acme) h BAD ON OF ye 27 fo. gsin 6 ds a g+{ y ds... “(zy AB AB Here the expressions J ap ‘| ap Pade A B denote the integrals indicated, taken from some chosen initial point on the line of integration up to the point A and the point B respectively. If the circuit be closed dI/dt vanishes by (11), that is the time-rate of change of the surface integral of spin is zero. Hquation (12) then vives 2()) ou gsind ae'+ () Si ds'=0, mre which of course is the direct consequence of (7) obtained ubove as (9). Prof. A. Gray: Notes on Hydrodynamics. Pi Equation (11) is Lord Kelvin’s well known theorem. But (12) puts the matter in a new way and brings into view the different causes of variation of I. The latter equation, (12), is an integral equation of motion corresponding to the differential equation 09’ Ts aol | ee islets (14) which has been obtained as (6) above. From this the whole motion of the fluid can be derived. 8. As has been noticed, if we take ds’ in the direction of the stream-line we obtain, since @=0, a ees oes ea 4a ee ee which is the equation of motion for the stream-line direction, and yields at once the so-called theorem of Bernoulli. If we take ds’ in the direction of the axis of spin, that is along what has been called a vortex-line, we obtain again Da CONE SEAN Bhs Vis NED ees Thus Bernoulli’s theorem is true also along a vortex-line. It is thus possible, in the case of steady motion, to draw through each point of the fluid a surface on which lie inter- secting stream-lines and vortex-lines, and for every point of which y has the same value. Let a normal be drawn to such a surface at any point, and dn denote a short step from the point along the normal. Then clearly we have, by (14), | at So ee iL! ae ee ee ee Yet nt If @ be the angle between the stream-line and the vortex- line which intersect at the foot of the normal, and w be the resultant regular velocity at that point, we have ® sin 6=@,,, and therefore Ox . to) ! Fn 170g in O=0. Pe ae rete Yk 3 a This equation is given in Lamb’s ‘ Hydrodynamics,’ § 164. It is the particular case of (14) in which 6=4$7, o.,.=o sin 4, and 9q'/dt=0. The theorem of (14) is quite general. 8 Prof. A. Gray: Notes on Hydrodynamics. 9. If in (14) we take the step ds’ parallel to the three axes Ow, Oy, Oz in succession, and denote q’ for these direc- tions by u, v, w, we obtain the three equations of motion . ee = 2052 q sin Os Sear Ox, Ov fo) AE + 2ey 4 sin Bay= — 5 f - >. is) ow Ox cae 9 7 i en eae A ot + 2@57q Sin ce Oz J It is to be noticed that these three axes may be inclined at any angles, so that (18) are equations of motion for a system of any three axes. 10. If we assume that the axes are rectangular, and write, according to the usual notation, 7 oie GP Hoa gj Oe ees o¢— 9° aes ( oy O02 O2iae Of ) 108 and regard 2, 2y, 2¢ as vectors associated with the axes Ox, Oy, Oz respectively, the vector associated in the same way with any axis the direction cosines of which are 1, m, n is 2(lE+mn+n6). If now 1, m, n be the direction-cosines of the normal to the plane ds, ds', drawn outwards from the diagram (fig. 2), this vector is @s). Thus we can write (14) in the form or + 2q(lE + mn +n€) sin Fp 8 20 Of a3 a a Os! * th ey eee ( ) This form is much less compact than (14), but from it we obtain at once the usual form of equations (18) for the case of rectangular axes. Putting /=0, we get g’=u, qnsind=—v, gnsin@=w. Similar results are obtained by putting m=0, n=0, in succession. Thus (18) become for rectangular axes oe — 206 + 2wn= — = | Or —2wf + 2ut = — 9% te 0 Dun 2om =~ 2% | These equations are usually derived from the Hulerian equa- tions of motion for the rectangular axes. The forms, however, Prof. A. Gray: Notes on Hydrodynamics. 7 of (18) are more compact and more general, since they are applicable to any system of three axes. Moreover, they can be written down at once from (14) which it is easy to remember. 11. From (18) we might deduce v. Helmholtz’s vortex- motion equations, but they are perhaps most easily obtained in the manner indicated below. ‘They are introduced here as the discussion will be found to lead to what seems a new proof of the constancy of the moment (product of angular velocity by cross-section) of a vortex-filament. By (1) above the equations of motion for the axes Oy, O= are dv , 0 _aV_12 } of 40s Oy poy | (22) ge" 0 OV" lop |. of 0s) Oz poe J These also hold whether the axes are at right angles or not. Taking the axes at right angles differentiate the first equa- tion with respect to z, and the second with respect to y, and subtract the first result from the second. We obtain easily dé 2 OU Ou. dou ; where @= ge On) Or dx dy Oz the “expansion” (time-rate of dilatation per unit of volume). Iwo similar equations hold for 7, and can be written down at once. It may be remarked that the right-hand side of (23) can be written as in the alternative equation Fone 48h 48". (28) Similar equations hold of course for the other axes. From the equation of continuity 2 +pO=0 we obtain by substitution for © in (23) and (23’) . ()= Pega nen (oOs | Sou Oy 4) Sot — = ao -—— —_— = = B = di\p/ pox poy ' pds plik et aeedl (24) 10 Prof. A. Gray: Notes on Hydrodynamics. The first of these is typical of the three equations of v. Helmholtz generalized for the case of a fluid of varying density. The alternative forms here shown are sometimes convenient. If we multiply the first of the equations typified by (23) [or (23') ] by & the second by 7, and the third by & and add, we get, since w= &+77+C’, dw ; Ou Ov Ow or, +e Bata e hae +oac - . . Bia where Ou/dc,...., denote differentiation of wu, v, w along the vortex-line, in the direction given by the cosines (&, 7, &)/a. The equation just found may be written dw E Ow” a ie 5 oC Ow Fe ak, one Boe ot ee (26) 12. Now the quantity on the right is the time-rate per unit length at which an element do of the vortex-line is increasing in length in the direction indicated by the cosines (~, n, O)/o. For du/dc.do, dru[dc.dco, Qu/da.do are the time-rates at which the projections of do on the axes Oz, Oy, Oz are lengthening. It is to be noticed very carefully that this is no¢ the unital rate of elongation of the element do of the vortex-line, for that is clearly 2 (Eu+ = oie w), 0 \@ @ @ which differs from E 0u.° 7 Ov | © 10 ® 0c! wdc oda by inclusion of the terms 0 (é 3 (7 SO: #8 uae )+e5c(2 )+%55(o) The expression on the right of (25) is thus the unital rate of elongation in the direction of the tangent to the vortex-line at the point considered. It is thus a component part of the dilatation of the fluid at that point, not following the fluid as it moves. But the dilatation © is the volume dila- tation also at a fixed point in space, and might be expressed equally well as the sum of a unital time-rate of elongation along the instantaneous direction of do, and a unital time- rate of expansion of area at right angles to that direction. As will be proved below, that areal expansion will, to the “Brae, A. Gray: Notes on Hydrodynamics. 11 first order at least of small quantities, be unaltered by asso- ciation with the fluid as it moves, since, to the degree of approximation stated, the effect of the displacement of the fluid will be simply to turn the area round through a small angle. Supposing then © to be expressed as has been suggested, we subtract the time-rate of unital elongation in the direction of the tangent to the vortex-filament from both sides of (25), and obtain Grote fe egal ie ee eee Ye where > is the unital time-rate of increase of area. If S be the cross-sectional area of the vortex-filament at the point considered, we have S=S 5, and therefore also oS+oS= 0, that is ®S =constant. The moment of the vortex-filament thus remains unaltered as the fluid moves. I have not hitherto seen this important result derived directly from equation (22). Moreover, the proof seems free from the objections brought by Stokes to the proof given by Cauchy, objections to which the proof given by v. Helmholtz is also open. 13. As to the point referred to above regarding the areal expansion at right angles to the tangent at P to the vortex- filament, let rectangular axes be so chosen at P that the axis of the filament is along Pz, and consider the expansion in the plane which at the beginning of an interval d¢ is at right angles to Pz. Take a distance APB=dw extending from — dx to +}dx, and another CPD extending from —43dy to +3dy. At A and B the component velocities of the fluid are jen OF ay _1dw ew! Eee ; U+ °) 504” Ut+ 5) 5a wW+ 9 en where the upper signs apply to A and the lower to B. Initially, then, the coordinates of A and B are —1 dz, 0, 0, and +3dz, 0,0, and after dt has elapsed these have become ae | ame . Lav ) (wo ) ¥ 5 du+(u—5 3" de) dt, (v5 oe de dt, wtes, ae dt. 12 Prof. A. Gray: Notes on Hydrodynamics. The projections of AB on the axes after dt has elapsed are therefore Ou Ons: OIE ike ae (1 5 Sdt), Sidedt, SP dedts and if small quantities of the second order be excluded the new length of AB is Olen a (1 i dt) Similarly we obtain for CD in its new position projections on the axes which can be written down from those for AB, and the length of CD has changed from dy to dy (1+ Sat). Further, it can be shown that to the degree of approxi- mation specified the angle ¢ between the new lines AB, CD (which intersect in the new position of P) is given by cos f= & + or at It follows that the former area dx dy has in its new position the value ou dv and the unital time-rate of areal expansion is ou ov Ox? Oy following the fluid, that is the statement made in § 12 is justified. . Some further notes regarding vortex-motion, dealing principally with surface and volume integrals, are reserved for another communication. ) be ey II. Notes on Hydrodynamics. By Professor A. Gray, P.R.S.* Il. Determination of Translational Velocity of a Vortex Ring of Small Cross-section. ie A VALUE for the velocity of displacement of a vortex ring in an unlimited fluid was given without proof in a note by Lord Kelvin appended to Professor Tait’s trans- lation of Helmholtz’s celebrated paper (Phil. Mag. 1867). A demonstration of the result was promised in the note, but, so far as I know, Lord Kelvin never published it. Several investigations have since been published with results differing slightly from Lord Kelvin’s value, which, however, has been confirmed by Hicks (Phil. Trans, 176) and by Lamb (Hydrodynamics, 3rd ed. p. 227). The following direct and elementary proof (in which no use is made of elliptic in- tegrals as such) may possibly be of interest. It is well known that the velocity at any point P ina frictionless incompressible fluid, in which exists a single vortex filament of any form, may be found in the following manner. Let « be the strength of the vortex, that is double the (constant) product of the elemental angular velocity in the filament at any point by the area of cross-section there, 7 the distance of the point P considered from an element H, of the filament, of length ds, and @ the complement of the angle between the directions of the element E and the line HP. The velocity dg at P due to the element EH may then be taken as given by the equation we ds.cos 6 graye Bure es) fi) 3 3K The direction of 6g is at right angles to the plane determined by the element E and the line EP, and the flow is towards the side of the plane specified by the rule given below. This rule applied to all the elements of the filament, gives the velocity at P as the resultant of all the vectors dg given by the elements composing the complete filament. Of course there might be added to the right hand side of (1) any term of proper dimensions for which the integration round the closed filament gives a zero vector. The theorem here stated is the analogue of that by which in electromagnetism the magnetic force at any point due to a linear cireuit may be found. There, if y be the current in * Communicated by the Author. 14 Prof. A. Gray: Notes on Hydrodynamies. the circuit of which an element Ei has length ds, and dI the field-intensity, Sl =y cos 0%, oe a) and 81 is at right angles to the plane determined by Hand P. The resultant intensity at P is the resultant of the complex of vectors 6I given by the elements composing the circuit. The Amperean rule for the direction of 61 is well known: that for the direction of dg is simpler. Imagine a closed path drawn round the element EK, in such a manner that the projection of the path on any plane dves not cross itself, and at Hi let the path lie along the normal specified. Let a point move round the path in the direction of the rotation at E: the direction of motion at P is the direction of dq. 2. Now imagine a circular vortex-filament of infinitesimal cross-section to exist alone in an unlimited incompressible fluid. Ifa point P be taken in the plane of the filament, the velocity (q) there at right angles to the plane is given by An ds p= 0058s eee where 7 is the distance of P from the element EH, of length ds, and @ is the complement of the angle between ds and the line HP. From this we can obtain an approximate evaluation of the surface integral of flow through a circle coaxial with the filament and differing only slightly in radius. Let the outer circle, of radius a in the diagram, represent the filament, and the inner circle, of radius a—«, represent the coaxial circle, in the same plane. The angle CEB is the angle @ as defined above, and so for the flow, dy say, through an element of area rd@ dr (HP=r) at P, due to the element HE of length ds, we obtain __ «C080 i dsd0 dr...» ))—e Aqrr Thus, if we suppose @ to vary from 0, when HB is along HC, the upper limit is sin-'{(a—2)/a}. We call this 0;. The limits of r are the two roots of the equation 1’ —2arcos0+a?—(a—a2)?=0. . . . (5) Approximately, these roots are 2acos@, x/cos@; a closer Prof. A. Gray: Notes on Hydrodynamics. 15 approximation gives 2a cos @—./cos 0, z/cos@. If we confine ourselves to the rougher approximation we get easily from the equation K 2a c0s@ [| 81 ags 8 = — 24 asf ) ——dé@dr, .. (6 ‘ 4a z/cos @ 0 _ ( ) for the total flow, the result x=«a(log~ —2), * ony ety rsp) a (7) The more exact value of the larger root of (5) might have been used without added difficulty, but a more accurate solution can easily be derived from (7) in another way. The order of approximation so far adopted takes that part of the flow through the filament, which escapes passing through the coaxial circle, as equal to that which might be computed by taking the filament as straight. Of course, if the filament is infinitely thin, this part is infinite, but the infinity is avoided in any actual case by taking the cross- section as finite though very small. The flow which passes outside the smaller circle may then be written xa log (a/e), where eis a very small quantity. The term xaloge also appears in the flow through the circle of radius a, so that for the smaller circle has the value stated in (7). Now let the smaller circle be moved out of the plane of the larger through a small distance y while remaining coaxial with the latter. If, then, ¢ be the shortest distance (=,/2’+y’) between two points, one on the filament, the other on the circle, the additional flow which escapes passing through the circle of radius a—wis xa(loge—logz). Hence, to the same order of approximation as before x=na(Iog" —2) 5h eee sf ea 3. Of course c involves a, but any attempt to calculate from (8) the axial component of velocity at the circumference of the circle of radius a—x would lead to an erroneous result, in consequence of our having neglected quantities of the first order in « We can now obtain, however, a closer approximation to y by writing, as was done by Maxwell (Electricity and Magnetism, § 705) for the electromagnetic analogue, y=n(Alog +B), . SAL MLO and then determining A and B from the physical fact. that 16 Prot. A. Gray: Notes on Hydrodynamics. the flow through the circle of radius a~# due to a vortex filament of strength and coinciding with the larger circle, is equal to the flow through the latter circle due to a vortex of the same strength coinciding with the smaller circle. We assume therefore A=a+A,e+...., B==—2a+B.2+.... 2 Since the vortex filament (as it was in the case considered by Lord Kelvin) is to be taken of small though finite cross- section, we need not carry the calculation beyond terms involving the first power of the ratio zla. No first power of y, or indeed any odd power, can enter, since the value of x cannot be altered by changing the sign of ». We substitute then in (9), with the values of A and B from (10), a—w for a and —z for 2, and obtain lz 8a 1 x) ; yana | (1-5 5)log-F -245 =}. . Mec), 4. We now consider a vortex ring of small circular cross- section (radius 7) made up of thin coaxial vortex filaments, each of the same strength per unit area of section, and calculate the flow through the circle, of radius a, which forms the circular axis of the anchor ring surface of the assemblage of filaments. Thus for the different filaments c varies from 0 to *. To find this axial flow, we calculate the rate of change of x when the ciicle through which the flow is taken is widened, while the filament producing it remains fixed. That is, we have to differentiate + with respect to x while R=a—vz remains unchanged. Thus, substituting R+< for a in (11) we get 8(R+.2) 3 NK | (R+ $2) log seer —2R— rh } . (3 and therefore aN 2 log a Oa ie = Lk GO Me sabi 9) 1s The axial component dv of velocity at any point of the cirele of radius a is this divided by 27ra. Thus ee 28a K fe) Da K x on= gea(loe; —!)—geala* a) © eg ee Prof. A. Gray: Notes on Hydrodynamics. 17 If now we write «=2 cdc d0, and integrate from 6=0 to 6=27 (remembering that cos 0=a/c), and fromc=0 to c=r, so as to find the effect of all the filaments in the ring, we obtain without difficulty (14) Now, for certain reasons which we do not discuss here, the circular axis of the vortex ring is a coaxial circle of radius Ry given by (15) and of distance &), from any chosen point on the axis of the system given by . KR? b= See sees es ae He The distance &) is thus equal to the axial distance of the circular axis of the anchor ring from the same chosen origin, that is the two circular axes specified lie in the same plane. The value of Ry is equal to the radius of gyration of a uniform circular lamina of radius r about an axis in its plane and at distance a from its centre. We have theretore Ro=/a? +11 =at 8 a” ° > ° . > » (16) approximately. In consequence of the spin @ the rate of advance of the circular axis of the vortex ring is less than the value of v above obtained by $w7?/a. Thus we obtain finally, writing « for the strength 2w77r* of the whole ring, = (log —z). 2s. 7) dt Ara r 4 the value given by Lord Kelvin. 5. Any other point in the plane of the circular axis of the vortex and near that axis might have been taken instead of a point on the circular axis of the anchor-ring surface for the specification of the cirele at which the axial velocity is calculated. In this case, however, it will be found more convenient for the sake of the integrations to calculate first the whole flow through the circle chosen due to the complete vortex ring, and then determine the velocity sought by differentiation. Phil. Mag. 8. 6. Vol. 28. No. 163. July 1914. C 18 Prof. W. M. Thornton on the If & be the radius of the circle chosen y can be expressed in terms of a, &,and r. Differentiation with respect to & and division by 27 give, to the degree of approximation adopted in this investigation, for the axial velocity fo oe ee lay? | — ES e 20" (; loo +7) where h=a—é&. Ifh=r, so that the circle chosen is the smallest, parallel to the circular axis, that can be taken on the anchor-ring, this becomes From this result the same equation (17) as before for the velocity of the circular axis of the vortex ring can be obtained. Innellan, April, 1914. III. The Lost Pressure in Gaseous Explosions. By Prof. W. M. Tsornton, D.Se., D.Eng., Armstrong College, Newcastle-on- Tyne *. (1) The “ suppression of heat” caused by the forces of cohesion in molecular jormation. 1 Vea a quantity of heat Q is given to unit mass of a gaseous mixture by slow external heating at con- stant volume, the rise of pressure is proportional to the increase in the absolute temperature and can therefore be calculated when the appropriate specific heat C, is known. If, however, the heat is derived from the explosion of the mixture the maximum pressure obtained experimentally is about half of that calculated from the heat developed, allowing for change of volume in combustion and with values of Q and ©, derived from experiments under steady conditions. Four suggestions have been made to account for this difference—(1) that there is rapid cooling by the cylinder walls (Hirn) ; (2) that there is dissociation of the gases formed (Bunsen) ; (3) that the specific heats rise greatly with the temperature (Mallard and Le Chatelier); (4) that there is ‘‘afterburning” (Clerk). Evidence has been given that the first three do not explain all the observed difference. Clerk’s argument is largely based on the fact that in certain cases the pressure in the * Communicated by the Author. Lost Pressure in Gaseous Explosions. Lg explosion cylinder remains constant for long intervals. This persistence of pressure can, however, be possibly explained, as shown later, by consideration of the transfer of energy in the molecules of the products of combustion, after the act of coherence by which new molecules are formed. In slow combustion a certain portion of the total energy of combustion is given to the products as heat, the amount depending eventually upon the equipartition of energy. In almost all physical processes into which consideration of equipartition enters, the transfer of energy is from translation to rotation or vibration. But in the act of combustion forces (of cohesion) are suddenly introduced which materially influence the method of transfer of molecular energy from one form to another. There cannot he a rise in the number of degrees of freedom sufficient to account for the deficiency in translational energy. In several cases a rise from 7 to 12 degrees each sharing the energy equally is necessary, and this does not occur, for it has been shown by Hopkinson * and David { that the energy radiated up to the point of reaching the maximum pressure is only 3 per cent. of the total energy of combustion, and further, the increase in the possible degrees of rotation is at most three. (2) Simple case leading to a mean efficiency of explosion of +. The mechanics of two colliding and cohering spheres show, however, that in such a case there may be a rise in the rotational molecular energy at the expense of the trans- lational, and this of an order which may account in part for the lost pressure. In the explosion of a hydrocarbon gas carbon monoxide is probably first formed ft, and the atomic weight of carbon is 12 and of oxygen 16. Take for simplicity of treatment two spheres of equal mass moving into union with equal velocity v, having angular velocity w, and having equal translational and rotational energies before collision. If their approach is in opposite directions in the same straight line, the trans- Jational energy is all converted into vibrational energy at the moment of collision, the total rotational energy either remaining the same as before contact, or if checked also adding to the latter. * Proc. Roy. Soc., A. vol. Ixxix. p. 138. + Phil. Trans. Roy. Soc., A. vol. cexi. p. 375. { H. B. Dixon, Phil, Trans. elxxxiy. p. 97 (1893), C2 20 Prof. W. M. Thornton on the Tf they approach out of line in parallel paths and cohere, the whole of their translational energy is converted into rotational energy (an irréversible process in combustion), and the molecule spins about the point of contact. In the case of oblique incidence we may resolve into two com- ponents and consider the energies before and after contact. Fig. 1. Before contact the total energy of each sphere is $(mv? + Iw), I being its moment of inertia. The translational energy of A parallel to B is $mv?sin?@ ; so that in union each sphere loses by this amount, for this is the component causing rotation about the point of cohesion. The total rotational energy of the two before collision is Tw’, after collision it may reach Iw?+ mv? sin? @. The mean value of sin?@ is $3 thus, since Iw? was taken to be equal to mv?, the rotational energy of the doublet formed is at most 14 times the rota- tional energy of its component before collision. The translational energy is before collision mv?; afterwards it is 4mv?(1—sin? @) +4mv’* cos? 8, for B loses a part of its translational energy in combining with A and the vertical component of the latter is unchanged. This expression is equal to mz*cos?@, and in the mean to 4mv?. Thus the ratio of the total translational energy after collision (to which the pressure of the gas is proportional), to that immediately before collision is }; in other words, the resulting pressure is only one half of that which would be obtained if no new molecules were formed. The argument holds for com- bining atoms of unequal mass, but with the same energy of translation, as for example hydrogen and oxygen in complete mixture. This excess of rotational over translation energy is not permanent and is more or less rapidly equalized to pressure and radiation; but there is no experimental evidence other than that derived from the velocity of an explosion wave, to show its actual duration. The conditions in a travelling explosion wave are somewhat different from the explosion at constant volume under consideration. Dixon has shown Lost Pressure in Gaseous Explosions. 21 that the velocity of a travelling wave corresponds to a temperature in the wave-front twice that derived from the heat liberated by slow combustion. Chapman has calculated the specific heats of the products from the wave velocity, and has obtained values nearly twice as high as those at low temperatures. Berthelot has remarked that the pressure indicated by any moving mechanism is probably not that in the wave-front but on: corresponding to ordinary com- bustion. If, however, the translational energy is suppressed for a moment by one half in the act of explosion, as shown above, the observed maximum pressures are the theoretical maxima and the difficulty of reconciling theory and observation is removed. (3) Experimental values of the Explosion Pressure ratio approximate to $. The ratio of the observed to the theoretical maxima, calculated on the assumption that all the energy of slow combustion is converted into translational energy in explosion, closely approaches one half. In the case of hydrogen and air the observed and cal- culated values are as follows *:— TABLE I, Maximum Pressure of Explosion in lbs. Percentage of Gas per square inch. in Air. Observed. | Calculated. Ratio. NIL ic dvevcacecer sk 41 88'3 ‘465 ee 68 124 549 5 oe a 80 176 455 | Meat. <. hoods: 489 In two cases of coal-gas and air Dr. Clerk obtained the following values f. * D, Clerk, ‘The Gas, Petrol and Oil Engine,’ vol. i. p. 105, ¥ Loe. cit. p. 136. 72 Prof. W. M. Thornton on the TABLE II, Maximum Pressure. Percentage of Gas | in Air. Observed. Calculated. Ratio. 7. i, eG eres iL | 6D On 7A MS: AYA SNE ASR SA 40 55 89°5 105 “447 "523 Den MECCONCR EER cg 51:5 67 96 110 535 602 Ha Gereeieo Bias et Cae 60 75 | 103 118 ‘582 645 Bae) Men eR Ae nae 61 76 =| 112 127 545. | +598 BO SOUR e ese are te setae 78 93 | 134 149 O72 "625 LG hia a RON a Breen 87 102 | 168 183 “517 “557 | FLORA, Oe IR i i ae 20 105 | 192 209 ‘468 "505 Mean...| °523 57) I. Glasgow Gas. II. Oldham Gas. These are the only recorded values at different percentages of gas and air readily available. Berthelot & Vielle (Annales de Chimie et de Physique, 6° série, tom. iv. pp. 3-90) obtained the following maximum pressures of explosion (Table III.). Berthelot had previously calculated* the theoretical maxima with values of the specific heats of the products which are certainly low, and therefore give too high a calculated maximum. The mixtures here are in every case those for perfect combustion. From an extended series of measurements on acetylene, Grover found ratios of observed to calculated maximum between 0°42 and 0°73. SBerthelot’s ratio for this gas is probably low, but retaining it the mean of all the ratios in Tables I., IL. and ILI. is 0°502. (4) Relation of efficiency of eaplosion to percentage of combustible gas in miature. The simple case considered cannot do more than suggest the chief process in gaseous explosion, for there is the addition of a second atom of oxygen to form carbon dioxide. It is to be noted, however, that this addition does not affect the ratio between translational and rotational energies so much as the first combination. Further, in Table II. the * M. Berthelot, ‘Explosives and their power. ‘Trans. by Hake and MacNab, pp. 387 and 543. Lost Pressure in Gaseous Explosions. 23 Tase III. Maximum Pressure in atmospheres. Combustible Gas in Oxygen. Observed. Calculated. Ratio. PEVOYORED 5.00.5. ..000006 9°8 20 0°49 Carbon monoxide...... 1071 24 0°42 WyaMOP OM «2... .0,0.5.00- 21:0 51 0°41 PMECOEVICHC 7.2. .00s000s~+- 15°3 45°5 0°34 Wihiyleme’ .........se0+ 1671 42 0°38 PANE, onan wer noeaern os 16:1 38 0°42 POR OAMG, ess ceces nee see 16°3 34 0-48 | | Mean 1 ie 0:42 PER CENT OF GAS IN AIR more inflammable mixtures have the higher ratios, and the variation of the efficiency of explosion with percentage of gas appears to be ina regular manner. In fig. 2 the ratios in Table II. are set down graphically, and the curves are seen to have the same shape throughout, I. being for Glasgow gas, II. for Oldham gas. 24 Prof. W. M. Thornton on the When two atoms combine to form a molecule their energy of combination is shared with surrounding inert molecules, and the greater the proportion of inert gas the more com- pletely the observed maximum should approach a maximum calculated on an energy basis only, since whatever form the energy of combination may for the moment take, it must be shared equally with the neighbouring molecules and in the end the translational or pressure energy is equalized. Thus the ratio of observed to the calculated maximum pressure would approach unity in the weakest mixtures if combination were not checked by the insufficiency of energy, set free by the molecules which are formed, to raise the whole mass of gas to a temperature such that all the combustible portions may unite. At the upper and lower limits of inflammability the ratio expressing the ‘‘ efficiency ” of explosion is of course zero. For coal-gas these limits are usually 29 and 6 per cent. Straight lines drawn from the upper limit through the curves cut the vertical axis at 0°94 and 0°98. [In the case of hydrogen, for which only three points are available, a line from the upper limit at 70 per cent. through two of the points cuts the axis at 0°78.| From this it is seen that the efficiency of explosion of the richer mixtures decreases to zero at a uniform rate as the percentage of gas is raised. In weaker mixtures, diluted with air, every atom of com- bustible gas enters into combination and shares its energy with the residue of the air. Thus the ratio of rotational to translational energy is not at any moment abnormally great for the whole volume, and the percentage of pressure developed is higher than if there were no nitrogen present. On the other hand, in richer mixtures with excess of com- bustible gas, all the oxygen is taken up but less combustible gas, and at the upper limit none of the latter. The efficiency of explosion should, therefore, always diminish in mixtures above the point of perfect combustion. Let N be the number of combustible gas molecules entering into combination in unit volume of mixture, n of ‘“‘inert ’? molecules. The ratio 7 of Table II. and the figure is that of the translational energy of the whole mixture to the energy of combustion in it. The translational energy is directly proportional both to N, the combustion of which gives rise to the heat, and to x which helps to transfer it as pressure to the walls, and therefore to their product. As the percentage is varied the sum N-++-n=M say remains constant, and Nn=N(M—N). Thus we may write the translational Lost Pressure in Gaseous Explosions. 25 energy in terms of N=aN—dN?’ and the efficiency ratio = alien? = Wo uae where gq is the heat of combustion of a single molecule. We should then have A 7 = A—BN, Ms and this is the equation of the line CD in the figure 2 obtained from the experimental results. (5) General relation of maximum pressure of explosion to percentages of combustible gas. Let the percentage of combustible gas in a mixture be represented by lengths from O to M, M being 100 per cent. Set up vertically at M a line MR to represent to any scale unit volume of combustible gas. At O set up OS to repre- sent unit volume of oxygen. Join OR and MS. The percentage P in the mixture giving perfect combustion is at that point at which the ratio of the ordinates PQ/PT is that of the volumes required for perfect combustion. P being a fixed point the lengths of OS and RM are decided by taking PQ and PT to a convenient scale in the correct ratio, and producing MQ to 8, OT to R. Fig. 3. R s a ° ea — U M In the case of methane for example, which requires twice its volume of oxygen for perfect combustion, PQ=2PT. Join OQ, then the ordinates of this line represent to a certain scale the number of combustible units* in the mixture below the point of perfect combustion, the ordinates of QM of those above this point. Let U and L be the upper and lower limits of inflammability. At the lower limit there are FL combustible units, and they fail to ignite the mixture because of the cooling influence of the mass of inert gas ; at * The combustible or explosive unit is defined as the aggregate of one molecule of combustible gas and the oxygen molecules required for its complete combustion. 26 Prof. W. M. Thornton on the the upper limit there are GU units, but the cooling or retarding influence of the gas is greater than at the lower limits and ignition again fails. The ratios Wale SDR eva FL OF VaGw are those of gas to air for perfect combustion. Now the heat set free in combustion is directly propor- tional to the number of combustible units in the mixture, except at the limits where the inflammation suddenly ceases. We may then write, for rich mixtures, from P to M, Q = Q.(1—aN), where Qc. is the heat of combustion of unit volume of the perfect mixture. For the part LP, Q ca Q.oN. The maximum pressure calculated from the heat liberated is Pe Par (14 a) where p, is the initial pressure, vo, v; the volumes before and after explosion, Q the total heat a combustion, and « the coeflicient of expansion of the gas. Substituting the above values for Q we have, since for the same gas v/v 18 constant, above the point of perfect Pp 2 p= ve Q.(1— ee combustion P : below P: D.==Cpi (1+ Lee By definition 7 = p obs. [P,, so that (a) above P: Pov. = cpin{1+ki—aN)}; and 7 has been found to be of the form A—BN, therefore in rich mixtures P obs. = ep(A—BN){1+4(1—aN)}, which may be written Pp obs. —= pi(Ay— BN + C,N?). Lost Pressure in Gaseous Explosions. 27 This is the shape of curve | (fig. 4), found experimentally by Grover for coal-gas and acetylene, in weak mixtures with | Pobs. a UE CET RARE ID CEE FERED initial compression *, that is, with the number of combustible molecules per unit volume N increased mechanically in the ratio of compression. (6) below P: P obs. = P\(A—BN)A(1+6N), — P1 (A, =e BLN —= C,N?). Fig. 4. This agrees with the shape of curve 2 (fig. 4), obtained in explosions at initial atmospheric pressure +. The point P appears, however, to be passed through without a sudden change in the curve. The agreement shows that by the use of the expression taken for the explosion efficiency 7 = A—BN, and the conception of the unit of combustion, the two types of curve expressing the experimental relation of maximum explosion pressure to percentage of combustible gas may be explained. The peculiar feature of the curves connecting the observed velocities of explosion with the percentage of combustible gas is that they consist, for the most part, of two straight lines forming a triangle upon the axis of N with the vertex at or near the percentage of complete combustion, and the base coinciding with the range of inflammability. The straightness of the rising and falling sides of this can be explained if the velocity falls from a maximum in proportion * Clerk, ‘The Gas, Petrol and Oil Engine,’ vol. i. p. 164, fig. 58. + Loc. cit. figs. 46, 49. 28 Prof. W. M. Thornton on the to the decrease in the number of combustible units from their maximum at P, the point of perfect combustion. PQ represents to a certain scale the maximum velocity of explosion; the ordinates of LQ, QU the velocities of explosion as the percentage of gas is increased from the lower to the upper limit of inflammability. — (6) Influence of initial compression on the efficiency of explosion. The expression for the efficiency may be also used to show how compression improves the efficiency of combustion. Other conditions remaining the same, doubling the pressure doubles the number of combustible units in a given volume. ‘This is represented in fig. 6 by extending the upper limit. OU, is the upper limit at atmospheric pressure, OU, at two atmospheres, OU; at three, and so on. The lines corresponding to CD in fig. 2 are CU;, CU., OUs.... It is well known that mixtures which are so weak that they cannot be ignited at atmospheric pressure are inflammable when compressed, so that N at the lower limit does not Lost Pressure in Gaseous Explosions. 29 advance in the same ratio as the upper limit. If the lower limit advanced at the same rate as the upper the efficiency would not be changed by compression. The efficiency of explosion is seen to be raised by initial compression, and it should continue to rise indefinitely, a oint of practical interest in internal combustion engines, which has led to discussion. (7) Summary. The total action in a gaseous explosion is a multiple of that which occurs in the formation of a single molecule. From the consideration of a diatomic case a value of 4 was obtained for the ratio of translational energies before and after formation. Kxperiment shows that this ratio varies in a definite manner and that the mean of its values over the working range in coal-gas, and in many other mixtures giving perfect combustion, also approaches 3. The suggestion now made is that the ‘‘ suppressed heat” in gaseous explosions may be explained by the influence of forees of cohesion which come into action at the moment of “contact” of two combining atoms. It would explain (1) the cause of the lost pressure; (2) the variation of the efficiency of combustion with strength of mixture ; (3) the shape of the maximum pressure-percentage curves ; (4) the differences between the explosion efficiency of gases having different limits of inflammability ; (5) the influence of initial compression in raising efficiency of explosion, as a consequence of the limiting shape of the efficiency curves. After explosion the molecular translational energy of the products of combustion in a closed vessel would fall more slowly than in simple cooling of a hot gas in which equi- partition is already established. This would have the same influence on the pressure as after-burning, and may be the cause, at least in part, of the maintained pressure observed by Clerk in rich mixtures (in which, as shown above, the pressure efficiency should be lowest), and of the prolonged radiation observed by Hopkinson after the flame stage has ceased. Leone IV. Notes on the Motion of Viscous Liquids in Channels. By J. PRoUDMAN *. 1. JN a recent communication to this Journal tT Messrs. Deeley and Parr remark that the conditions of the steady flow of a viscous liquid in a parabolic channel, under a constant force parallel to the length of the channel, have not yet been ascertained. It is implied that the results might be of interest in connexion with the motion of glaciers. In the present communication the problem is solved for the special case in which the free surface of the liquid passes through the focus of the parabolic section, and also for a particular triangular section. Some remarks are also added in connexion with the mathematical expansions used. The general problem { for a channel of any section may be reduced to that of finding a function y which satisfies OX) ron we Se ae 2) n.d over the section, which vanishes over the sides of the section, and for which 0yv/Qn=0 over the free surface. Here a, y are rectangular Cartesian coordinates in the plane of the section, and Q/dn denotes differentiation along the normal to the free surface. The velocity of the liquid, which is parallel to the length of the channel, is given by Py/2u, where P is the pressure gradient along the channel, and wp is the coefficient of viscosity. In applications, the function F =|\xd S, where the integral is taken over the area of the section, is required. Particular Parabolic Section. 2. For convenience, take the length of the latus-rectum of the parabola to be 477”. Then if we take polar coordinates y, 0, having for pole the focus 8, and for initial line the axis * Communicated by the Author. + “The Hintereis Glacier,” Phil. Mag. (6) xxvii. p. 153 (1914). t See Lamb, Hydrodynamics, 3rd ed., p. 545. 4 Motion of Viscous Liquids in Channels. 31 q __ SA, the equation of the parabola will be 7 cos i0=7, and that of the latus-rectum 9 6?=177”. Fig. 1. 2 = De S bo — eee pena | ' 1 ' ‘ ' | 1 ! fas} non. tt (as os ohms WT 1 a. LL ee Rae ---. € S| A I t 1 1 1 t ) 1 1 eee eS 2S Foe eae U Let us take E=7% cos 10, n=r2 sin 10, so that we have a conformal transformation if E, n be regarded as Cartesian coordinates in another plane, the parabola trans- forming into £=7, and the latus-rectum into =r. The correspondence is shown in figs. 1 and 2, where corresponding points are similarly lettered. rh Since O(a y) Sg es LO a a vie 0(&, 7) oe) (2) the conditions to be satisfied by y become, with reference to fig. 2, Ce ee OE 52 + <= =o.) gic ge er ae Us (3) over the area L'SL, y=0 on E=7, dy/9E=9 ae and Ox/d€=—Odx/d7 on E=—y. nek 32 Mr. J. Proudman on the Instead of trying to solve this problem directly, let us determine a function which satisfies (3) over the area of the square bounded by €=+7, n=+7, and which vanishes on the sides of this square. The determination of such a fune- tion is known to be unique, and from considerations of symmetry we see that over the triangle L’SL it will be the function we. require. Now 4(n?-+ &)(n®@—1")— & An cosh (n+4)£ cos (n+4)m, (4) where A, is a constant, satisfies (3) and vanishes over n=+7. We shall see that we can choose the constants An so that it will vanish on €= +7. From Fourier’s theorem, or otherwise, we have 9 48 (—1 i T—7? = a toad $08 (n-+4)n, for —7 < n < 7, by which we see that if we take (— 1 yn An cosh (n+4)r=327 (n+4)”” (4) will satisfy all the conditions for x. Thus, x= 4° +B) a?!) 390 & (—1)” cosh(n+4)é n=o0 (n+4)* cosh (n+4)r cos (n+4)m. (5) The value of y on SL, which gives the velocity on the free surface, is obtained by putting »=& in (5). Doing this, we obtain 2 (—1)" cosh(n+4 . A(t — £8) 320 & AO IEEE cos (nt dE, ©) n=0 in which € is connected with the distance r from the focus, by €=173/ V2. For the flux of liquid through the channel we require the function F=4 {| (+7) dé dn, taken over the area of the triangle L’SL (fig. 2), or, again from symmetry, taken over the area of the square SL. The integration is straightforward, the series for y being uni- formly convergent over the area, and we obtain FE qr® 9% 1 1 ieee oS ee 1 Se Motion of Viscous Liquids in Channels. 33 or, since j } < 7 = CEES rae F 7° © a: 1 (3 ae ha ea ae (n+4)7. If now we take the latus-rectum to be 4a instead of 47’, F will be multiplied by (a/z’)*, so that _ Eee > ae ale (n+4)r.. ~. (7) er a ang (O+2)° Particular Triangular Section. 3. The section is that in which one side of the channel is vertical and the other inclined at an angle 17 to it (fig. 3). Fig. 3. ' t i t { { ' ¢ t ‘ ! 4 i] ' ( ‘ ' wo s ee ween The solution for this case can be derived from an expression previously given*, but it is just as easy to verify directly that all the conditions of the problem are satisfied by nit 1 =e ‘OLS Ny ak p> ana ae 2 eee —2)—— 27 ean (Qn tla x {sinh (n+ 4)(2n—2+y) sin (n+4)(e+y) —sinh (n+4)(a+y) sin (n+4)(e#—y)}, . « . (8) the axes being as shown in fig. 3. The boundary conditions are that y=0 on w= and on e=—y, and that dy/dy=0 on y=0 ; but instead of the latter we take y=0 on w=y, again appealing to symmetry. We have taken OA=z for convenience. * Lond. Math. Soc., Records for March 18th, 1918. Phil. Mag. 8. 6. Vol. 28. No. 163. July 1914. D 34 Mr. J. Proudman on the The value of y on OA, which gives the velocity at the free surface, is 2 2 sinh (n+4)(7—2) sin (n+ $)a DO) 2 (n+4)?sinh(n+4)r 7° (9) and the function F, when OA is taken to be a instead of a, is given by F 1 i ae i Pica Diet a > ——;coth (nt+4)7.. . (10) a* Dar ip (n+ 2) Remarks on the Expansions. 4. The normal derivatives of (5) and (8) must vanish over SL and OA respectively. For (5) this gives us 2 2 (—1)* i | 2M eal ey iil Miao Pe Sal 8 °F p=0 (n +3) cosh (n+ 3)m x {cosh (n+9)Esin (n+3)E+ sinh (n + 3)Ecos(n+$)E}, (11) for —7<& ° 3 . (5) then ee not a—(h+1)n+1 I, —in,p-p—9= — n( p—B—h—1) T n(p—B—-h— 1) emia in coi If in (6) to h are assigned the values 0,1, 2,...., p—8B—2, and the resulting Integrals combined, we obtain — ga) II (a—yn+ 1) y=1 I she gs 1 a, p—B— = “pgn—a—lfinn 1 | \p—B-: g=1 ng il (p—B+y) uy M(x +1)? B-g9 v— bert Se I] (a-—yn+ 1) ie | o ne- =—p— i p—B- 1)! ; yr B—1)n- =a( go" +1)° * Communicated by the Author. 58 Dr. I. J. Schwatt: Note on The problem is therefore reduced to the finding of : da Py e e e s es 8 fe ge Bn al gn 1 ( ) The greatest value of 8 is c, and that of c is p—1, c being by hypothesis less than p. We must therefore consider the following two cases. Z wide (i.) B=c=p—l, then (8) becomes { ey? ae (9) | Elk aap and (ii.) B v ms aaa er ROS 2n(a@-+ 1) a 2e+- 1 TA n—2 Qn = | 20 cos— m(a+1)—2 cos vk ge =be[1— (= ein Wa Gare 1) ee peo r—2Qe — ar) (13) and q xrde al 1 ( Pe +] = — | cos m(a+1) { log (x? — 2x cos— a7+1) n { = e+ 1 n > 2 | ‘ peed VL ~2 log (2 sin a, ) | 2 { rm cos ti | +2 = 7 — tan! | ) 2etl me eee J pO hog (0+ 1), uo aa To find the Integral (10), we write 1 ye a yi Sr a ae 2: i - ~ j = can cia 1 hana . . . . . . L3 ee ror, | A ane ™— PBI )n—a (19) Multiplying both sides by w—7, and letting z=7,, we have A, —_— io] — 1 pics ao —n 1. eae (16) dh oe ye SOE lal eee ea | Multiplying both sides of ) by 2”, gives am m—1 : See AS MC A ag 2+1 Pe lea x=0 Differentiating both sides « times and letting #=0, we obtain | Bus: oul Pe V dupe = 24-1 ia Be en =a +x rsh It follows. that By=1, B,=B,=....=B,.1.=0, B,=—1, By,= +1, and in general B,,=(—1)*. 1 Ll 2 yea p—B—2 (—1)* : eater ie ON ates Yea AT 1) fe (a2 sp i) Ve k= Cine K=0 aP—B-«—1)n—« ( 8) And Foes | : | ie = | cos pd, 1) { log (a= 20 cos